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Theorem sucexeloni 7829
Description: If the successor of an ordinal number exists, it is an ordinal number. This variation of onsuc 7831 does not require ax-un 7755. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof shortened by BJ, 11-Jan-2025.)
Assertion
Ref Expression
sucexeloni ((𝐴 ∈ On ∧ suc 𝐴𝑉) → suc 𝐴 ∈ On)

Proof of Theorem sucexeloni
StepHypRef Expression
1 eloni 6394 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 ordsuci 7828 . . 3 (Ord 𝐴 → Ord suc 𝐴)
31, 2syl 17 . 2 (𝐴 ∈ On → Ord suc 𝐴)
4 elex 3501 . 2 (suc 𝐴𝑉 → suc 𝐴 ∈ V)
5 elong 6392 . . 3 (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
65biimparc 479 . 2 ((Ord suc 𝐴 ∧ suc 𝐴 ∈ V) → suc 𝐴 ∈ On)
73, 4, 6syl2an 596 1 ((𝐴 ∈ On ∧ suc 𝐴𝑉) → suc 𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  Vcvv 3480  Ord word 6383  Oncon0 6384  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by:  onsuc  7831  1on  8518  2on  8520
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